Séminaire BOURBAKI
62ème année, 2009-2010, no 1025

Juin 2010

THE ACC CONJECTURE FOR LOG CANONICAL THRESHOLDS[after de Fernex, Ein, Mustaţă, Kollár]
by Burt TOTAROINTRODUCTION
The minimal model conjecture asserts that every algebraic variety X not coveredby rational curves is birational to a projective variety Y which is negatively curved in
the sense that the first Chern class c1 (Y ) has degree ≤ 0 on all curves in Y , called aminimal model. The problem is fundamental to all kinds of classification problems inalgebraic geometry. The minimal model conjecture is known for all varieties of general
type [BCHM] and for all varieties of dimension at most 4, but the full conjecture
remains wide open. (Although the minimal model conjecture should be true in any
characteristic, the results mentioned work over a field of characteristic zero. For these
problems, we lose nothing by working over the complex numbers. Birkar [Birkarexist]
summarizes the known results on the minimal model conjecture in dimensions up to 5.)
Shokurov showed that the minimal model conjecture would follow from the ascending
chain condition for a certain invariant of singularities, the minimal log discrepancy(mld), together with a semicontinuity property for minimal log discrepancies[Shokurovmld]. The minimal log discrepancy is a rational number associated to a given
singularity, with bigger numbers corresponding to milder singularities. Shokurov’s
ACC conjecture says that the set of minimal log discrepancies of all singularities of agiven dimension is a subset of the real line that contains no infinite increasing sequence.
Let us say vaguely why this would imply the minimal model conjecture. Starting fromany projective variety, after contracting finitely many divisors, we can make it closer
and closer to a minimal model by a sequence of birational maps called flips. We get aminimal model if we can show that no infinite sequence of flips is possible. But eachflip improves the singularities as measured by minimal log discrepancies, and so the
ACC conjecture would imply that the sequence of flips terminates.
Shokurov also conjectured the ACC property for another invariant of singularities,
the log canonical threshold (lct). This conjecture should be a natural preliminary to
the ACC conjecture for minimal log discrepancies, since log canonical thresholds are
simpler than minimal log discrepancies and have been studied for a long time in singularity theory. In particular, ACC for log canonical thresholds is known in dimension at
most 3 while ACC for minimal log discrepancies is known only in dimension at most 2,

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by Alexeev [Alexeev1, Alexeev2]; moreover, ACC for minimal log discrepancies in dimension 3 would imply ACC for log canonical thresholds in dimension 4 [SB, Corollary1.10]. Log canonical thresholds have an elementary analytic definition: for an analytic
function f on Cn , the log canonical threshold of the hypersurface f = 0 at a point p
is the supremum of the real numbers s such that |f |−s is L2 near p. Estimates of log
canonical thresholds have a variety of applications in algebraic geometry, including theconstruction of Kähler-Einstein metrics on many Fano varieties [DK] and the proof of
non-rationality for many Fano varieties [DEMrigid].
The ACC conjecture for log canonical thresholds has some implication for the minimalmodel conjecture, albeit a limited one. By Birkar, the minimal model conjecture indimension n − 1 (for pairs (X, B) with KX + B pseudo-effective) implies the minimalmodel conjecture for pairs (X, B) of dimension n with KX + B effective [Birkarexist2].
If we also know ACC for log canonical thresholds on singular varieties of dimension n,then we can deduce termination of flips for pairs (X, B) of dimension n with KX + B
effective [BirkarACC]. Termination is a stronger statement than existence of minimalmodels (because it says that any sequence of flips will lead to a minimal model).
A recent advance is the proof of ACC for log canonical thresholds on smooth varietiesof any dimension by de Fernex, Ein, and Mustaţă [DEM]. Their method covers many
singular varieties as well, including quotient singularities and local complete intersections. In dimension 3, every terminal singularity is a quotient of a hypersurface
singularity by a finite group, and so de Fernex-Ein-Mustaţă’s methods reprove thegeneral ACC conjecture for log canonical thresholds in dimension 3. In dimensions atleast 4, there seems to be no hope of a comparably explicit description of terminal singularities. Nonetheless, de Fernex-Ein-Mustaţă’s work provides striking new evidence
for the ACC conjectures, suggesting that they form a plausible approach toward theminimal model conjecture. The final version of de Fernex-Ein-Mustaţă’s argument,
incorporating contributions by Kollár, is short and simple.
This exposition owes a lot to Kollár’s excellent survey of the ACC conjecture for log
canonical thresholds [KACC]. Thanks to Ofer Gabber for suggesting Corollary 1.7.

1. INTRODUCTION TO LOG CANONICAL THRESHOLDSDefinition 1.1. — Let f be a holomorphic function in a neighborhood of a pointp ∈ Cn . The log canonical threshold of f at p is the number c = lctp (f ) such that— |f |−s is L2 in a neighborhood of p for s < c, and— |f |−s is not L2 in a neighborhood of p for s > c.
Thus lctp (f ) = ∞ if f (p) 6= 0, and by convention lctp (0) = 0.
The log canonical threshold is a natural measure of the complexity of the zero setof f near p. It was considered by Atiyah [Atiyah] and Bernstein [Bernstein], andfurther explored by Arnold, Gusein-Zade and Varchenko [AGZV, v. 2, Section 13.1.5]

1025–03

and Kollár [Ksing]. The main textbook treatment of log canonical thresholds is in
Lazarsfeld’s book [Lazarsfeld, Section 9.3].
Shokurov had the idea that the set of all possible values of log canonical thresholds
in a given dimension should have special properties [Shokurovprob].
Definition 1.2. — Let HT n be the set of log canonical thresholds of all possible holomorphic functions of n variables vanishing at 0. That is,
HT n = {lct0 (f ) : f ∈ O0,Cn , f (0) = 0} ⊂ R.
The name HT n indicates that these are “hypersurface thresholds”. Indeed, it is easy
to see that lct0 (gf ) = lct0 (f ) for g(0) 6= 0, which says that lct0 (f ) only depends on thehypersurface {f = 0} near 0 ∈ Cn (considered with multiplicities). We get the same
set HT n if we allow f to run through all polynomials or all formal power series overany algebraically closed field of characteristic zero, using the algebraic definition of thelog canonical threshold in Section 3 [KACC, Section 5].
The function |z|−s is L2 on a neighborhood of the origin if and only if s < 1. It
follows that, for a holomorphic function f of one variable,
lctp (f ) =1.
multp (f )As a result,
HT 1 =
1 1
1, , , . . . , 0 .
2 3There is also a complete description of the set HT 2 , by Varchenko [Varchenko], [KSC,
Theorem 6.40], [KACC, Equation (15.5)]:
c1 + c2
HT 2 =: a1 , a2 , c1 , c2 ∈ N, c1 + c2 ≥ 1,c 1 c 2 + a1 c 2 + a2 c 1
a1 + c1 ≥ max{2, a2 }, a2 + c2 ≥ max{2, a1 } ∪ {0}.
The sets HT n are not known for n ≥ 3, and it may be unreasonable to expect an
explicit description. Nonetheless, they have remarkable properties. First, Atiyah used
resolution of singularities to prove:
Lemma 1.3. — All log canonical thresholds are rational and lie between 0 and 1. Thatis, HT n ⊂ Q ∩ [0, 1].
We will discuss de Fernex, Ein, and Mustaţă’s theorem:
Theorem 1.4 (ACC conjecture, smooth case). — For any n, there is no infinite
increasing sequence in HT n .

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By contrast, every rational number between 0 and 1 is the log canonical threshold ofa holomorphic function in some number of variables. Also, there are many decreasing
sequences of log canonical thresholds in a given dimension, as we see from the example[AGZV, v. 2, Section 13.3.5], [Ksing, Proposition 8.21]:
Lemma 1.5. —
lct0 (z1a1 + · · · + znan ) = minn 1
1o1, + · · · +.
a1an

Kollár described all limits of decreasing sequences of log canonical thresholds onsmooth varieties of dimension n:
Theorem 1.6 (Accumulation conjecture, smooth case). — The set of accumulation
points of HT n is HT n−1 − {1}.
In particular, HT n is a closed subset of the unit interval, although it is contained in
the rational numbers.
The ACC theorem implies that there is some n > 0 such that no log canonicalthreshold on a smooth n-dimensional variety is in (1 − n , 1) (the smooth case of theGap conjecture). The proofs are nonconstructive, and so there is no explicit lower
bound for n in general. There is a conjecture for the optimal value of n . Consider thesequence defined by cn+1 = c1 . . . cn + 1 starting with c1 = 2. It is called Euclid’s or
Sylvester’s sequence, and starts as:
2, 3, 7, 43, 1807, 3263443, 10650056950807, · · ·The definition of ci implies that
X1
1
1
=1−
=1−
.
ci
cn+1 − 1c 1 . . . cn
In particular, by Lemma 1.5,
1.
lct0 (z1c1 + · · · + zncn ) = 1 −cn+1 − 1
Kollár conjectured that this is the extreme case [KACC]. That is, the conjecture is thatno log canonical threshold in dimension n lies in the interval


1
1−
,1 .
cn+1 − 1This optimal Gap conjecture is known in dimension 3 (3 = 1/42) by Kollár [Ksing].
The set HT 3 is not known, but Kuwata determined HT 3 ∩ [5/6, 1] [Kuwata]. The ACC
conjecture for functions on singular 3-folds, which we have not formulated yet, wasproved by Alexeev [Alexeev1] and the Accumulation conjecture on singular 3-folds by
Mc Kernan-Prokhorov [MP]Finally, we can rephrase the ACC theorem, Theorem 1.4, as saying that the set
HT n turned upside down is well-ordered. Gabber pointed out that the Accumulation
theorem, Theorem 1.6, determines the order type of HT n by a simple induction:

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Corollary 1.7. — For each positive integer n, the set HT n turned upside down hasorder type ω n + 1.

2. FORMULA FOR THE LOG CANONICAL THRESHOLD
In this section, we give Atiyah’s formula for the log canonical threshold in terms ofan embedded resolution of the hypersurface {f = 0}.
Lemma 2.1. — If f (p) 6= 0, then lctp (f ) = ∞. If f (p) = 0, then 0 ≤ lctp (f ) ≤ 1.
Proof. The first claim is clear. So assume that f (p) = 0. As we said, for a one-variable
holomorphic function f (z), we have lctp (f (z)) = 1/multp (f ). In several variables, picka smooth point q near p on the hypersurface {f = 0}. We can choose local coordinates
near q such that f = (unit)z1m for some m. If a function is L2 on some neighborhoodof p, then it is L2 near some such point q, and so lctp (f ) ≤ lctq (f ) = 1/m.
In several variables, the multiplicity at 0 of a holomorphic function f means thesmallest degree of a nonzero term in the power series expansion of f at 0. In any
dimension, the log canonical threshold differs by a bounded factor from the multiplicity:
1n≤ lctp (f (z1 , . . . , zn )) ≤multp (f )multp (f )[KACC, (11.5), (20.1)]. The ACC property is trivially true for the invariant 1/multp (f )of functions f , as that only takes the values {1, 1/2, 1/3, . . .} ∪ {0}, but the ACC
property for log canonical thresholds is a deeper problem.
Let ω = dz1 ∧ · · · ∧ dzn . Then |f |−s is locally L2 if and only if on any compact set
K ⊂ U , the integral
Z(f f )−s ω ∧ ωKis finite. Let π : X → U be a proper bimeromorphic morphism. We can rewrite thisintegral asZZ−s((f ◦ π)(f ◦ π))−s π ∗ ω ∧ π ∗ ω.
(∗)(f f ) ω ∧ ω =K

π −1 (K)

Since X → U is proper, it suffices to show that the integral on the right is finite near
any given point of X.
By Hironaka, we can choose π : X → U to be an embedded resolution of the hypersurface f = 0. That means that X is smooth and the zero set of f ◦ π together withthe exceptional set of f form a normal crossing divisor. That is, at any point q ∈ X,
we can choose local coordinates x1 , . . . , xn such that
Y a (q)f ◦ π = (unit)
xi ii

1025–06andπ ∗ ω = (unit)

Y

e (q)

xi i dx1 ∧ · · · ∧ dxn ,iwhere ai (q) = multxi =0 (f ◦π) and ei (q) = multxi =0 Jac(π). Here Jac denotes the complex
Jacobian∂zi
Jac(π) = det.
∂xj
Then the integral (?) is finite near a point q ∈ X if and only if
Z YYZ
ei (q)−sai (q)(xi xi )dV = (const)(xi xi )ei (q)−sai (q) dxi ∧ dxi
ii

is finite. This holds if and only if ei (q) − sai (q) > −1 for every i, that is, whens < (ei (q) + 1)/ai (q). This gives the formula for the log canonical threshold:
1 + multE Jac(π)
lctp (f ) = min: for those E such that p ∈ π(E) .
multE (f ◦ π)
It follows in particular that the log canonical threshold of any holomorphic function f
is a rational number. We can rewrite the formula in the language of divisors. Let D be
the divisor of a holomorphic function f , meaning the linear combination with integerP
coefficients of the irreducible components of the hypersurface f = 0, D =di Di
where di is the multiplicity of f along Di . We can interpret the Jacobian of a resolution
π : X → U as describing the difference between the canonical divisors of X and U :X
KX = π ∗ (KU ) +ei Ei .
i∗

Also, π (D) means the divisor of the function f ◦ π, π ∗ (D) =ei + 1
lct0 (D) = min: ai > 0 .
aiPai Ei . In this notation,

Equivalently, in the terminology of minimal model theory, the log canonical thresholdof a divisor D on a variety X is the maximum real number c such that the pair (X, cD)
is log canonical [Lazarsfeld, Definition 9.3.9]. This definition makes sense on singularvarieties X as well, and the ACC conjecture and related conjectures make sense in thatgenerality.

3. THE LOG CANONICAL THRESHOLD OF AN IDEAL OF
FUNCTIONS
The definition of the log canonical threshold of a function was generalized to ideals
of functions by de Fernex, Ein, and Mustaţă [DEM]. They prove the ACC property inthe greater generality of ideals (on a smooth variety). This stronger statement turns
out to be easier to prove, because working with ideals makes it easier to concentrate on
the behavior of a function at a single point.

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Let a be a sheaf of ideals on an excellent noetherian scheme X over Q. (The main
example in this paper is the spectrum of a power series ring over a field of characteristic
zero.) By Hironaka, extended by Temkin [Hironaka, Kres, Temkin], there is a log
resolution Y → X of the ideal a. That is, Y → X is a proper birational morphism,
Y is regular, the union of the exceptional locus of Y → X with the inverse image ofthe zero set of a is a divisor ∪ Ei with simple normal crossings, and the ideal a · OY
P
is a principal ideal, of the form a · OY = OY (− ai Ei ) for some natural numbers ai .
That is, a · OY is the ideal of regular functions on Y which vanish to order at least ai
on the divisor Ei for each i. The image in X of a divisor Ei is also called the center of
Ei on X.
Throughout this paper, let k be an algebraically closed field of characteristic zero.
Definition 3.1. — Let a be an ideal contained in the maximal ideal mk of a power
series ring R = k[[x1 , . . . , xn ]]. Let π : Y → X = Spec R be a log resolution of a, andPPwrite a · OY = OY (− ai Ei ) and KY /X = KY ⊗ π ∗ (KX )∗ equal to OY ( i ei Ei ). Define
the log canonical threshold of the ideal a at the origin in X to be
ei + 1
lct0 (a) = min.
ai
The log canonical threshold of an ideal is independent of the choice of log resolution,by Theorem 9.2.18 in Lazarsfeld [Lazarsfeld].
For analytic functions f1 , . . . , fr which vanish at the origin in Cn , the log canonicalthreshold of the ideal (f1 , . . . , fr ) is equal to the supremum of the numbers s suchthat (max{kf1 |, . . . , |fr |})−s is L2 near the origin, by the same argument as for a single
function in Section 2.
For example, the ideal (x1 , . . . , xn ) in Cn has log canonical threshold at the origin
equal to n. (It suffices to blow Cn up at the origin; the exceptional divisor E1 has
e1 = n − 1.) More examples are given by the following version of the Thom-Sebastiani
lemma:
Lemma 3.2. — Let a ⊂ k[[x1 , . . . , xm ]] and b ⊂ k[[y1 , . . . , yn ]] be ideals in disjoint sets
of variables. Then the ideal c = (a, b) ⊂ k[[x1 , . . . , xm , y1 , . . . , yn ]] has
lct0 (c) = lct(a) + lct(b).
This follows by the same argument as for a single function [Ksing, Proposition 8.21].
For example, Lemma 3.2 gives the calculation
11+ ··· + .
lct0 (xa11 , . . . , xann ) =a1
an
For any ideal a in a power series ring k[[x1 , . . . , xn ]], it is immediate from the definition
that lct(am ) = lct(a)/m. Therefore it makes sense to define lct(aq ) for any positive real
number q, as lct(aq ) = lct(a)/q. This agrees with the definition of the log canonicalP
threshold if we think of the pullback of the object aq to Y as the R-divisor − qai Ei .

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Using pullbacks in this way, we can more generally define the log canonical thresholdof aq · br for any ideals a and b and any positive real numbers q and r.
Definition 3.3. — For a natural number n, let Tnsm be the set of all log canonicalthresholds of ideals a contained in the maximal ideal of k[[x1 , . . . , xn ]].
The set Tnsm is independent of the choice of algebraically closed field k of characteristic
zero [DM, Proposition 3.3]. It contains the set HT n of log canonical thresholds offunctions on a smooth n-fold, and so the ACC property for Tnsm will imply it for HT n .
The definition of log canonical thresholds shows that Tnsm is a subset of the rationals
contained in the interval [0, n].

4. APPROXIMATION OF THE LOG CANONICAL THRESHOLD
In this section we state a crucial lemma, Lemma 4.1, due to Kollár (in the main case ofprincipal ideals) [KACC, Proposition]. The lemma says that under certain conditions,the log canonical threshold of an ideal is not changed if we change the ideal by adding
terms of high degree.
Lemma 4.1. — Let a be an ideal in R = k[[x1 , . . . , xn ]] which is contained in the
maximal ideal mk . Suppose that the log canonical threshold of a is computed by somedivisor E on a log resolution of X = Spec R whose image in X is the origin. If b is an
ideal such that a + q = b + q, where q = {h ∈ k[[x1 , . . . , xn ]] : ordE (h) > ordE (a)}, thenlct(b) = lct(a).
The proof as simplified by de Fernex, Ein, and Mustaţă [DEM, Corollary 3.5] uses
only the Connectedness theorem of Shokurov and Kollár [Ksing, Theorem 7.4]. Here isthe case we need.
Theorem 4.2 (Connectedness theorem, smooth case). — Let g be a complex analytic
function on a neighborhood X of the origin in Cn . Let π : Y → X be a log resolutionPPof g. Write g · OY = OY (− ai Ei ) and KY /X = OY ( i ei Ei ). Let c be a positive
rational number. Then the union of the divisors Ei such that (ei + 1)/cai ≤ 1 isconnected in a neighborhood of π −1 (0).
The Connectedness theorem itself is an ingenious but quick consequence of the
Kawamata-Viehweg vanishing theorem, hence ultimately of the Kodaira vanishingtheorem. It is striking that the only deep ingredients of the proof of ACC for smoothvarieties are resolution of singularities and the Connectedness theorem, both of whichwere available by 1992.

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5. GENERIC LIMITS OF IDEALS
We define in this section a “weak limit” of an arbitrary sequence of ideals in a power
series ring, called the generic limit. The generic limit was first defined by de Fernex andMustaţă using ultraproducts [DM]. The construction was then simplified by Kollár touse only elementary algebraic geometry [KACC]; we follow the exposition by de Fernex,
Ein, and Mustaţă [DEM]. The generic limit shows that the limit of a sequence of log
canonical thresholds, arising from an arbitrary sequence of ideals, must be equal to thelog canonical threshold of some other ideal.
Let R = k[[x1 , . . . , xn ]] be the ring of formal power series over an algebraically closed
field k, and let mk be its maximal ideal (x1 , . . . , xn ). For a field extension k ⊂ K, define
RK to mean K[[x1 , . . . , xn ]] and mK = m · RK .
For each natural number d, we can identify ideals in R/md with ideals in R that
contain md . Let Hd be the Hilbert scheme of ideals in R/md , which is a closed subscheme of a finite union of Grassmannians (the Grassmannian of linear subspaces ofcodimension j, for each 0 ≤ j ≤ dimk (R/md )). Mapping an ideal in R/md to its image
in R/md−1 gives a surjection td : Hd (k) → Hd−1 (k), which is not a morphism of schemes.
But it is a morphism on the subset of Hd (k) corresponding to ideals in R/md whose
image in R/md−1 has a given codimension. Therefore, for each closed subvariety Z
of Hd (note that we understand a subvariety to be irreducible), td induces a rational
map Z 99K Hd−1 .
Let a0 , a1 , . . . be a sequence of ideals in R. I claim that we can choose a sequence of
closed subvarieties Zd ⊂ Hd such that: (1) td induces a dominant rational map from
Zd to Zd−1 for each d ≥ 0, and (2) for each d there are infinitely many i such that theideal ai + md corresponds to a point in Zd , and Zd is minimal among subvarieties of Hdwith this property.
Indeed, suppose inductively that we have constructed Z0 . . . , Zd−1 with properties(1) and (2). In particular, there is an infinite set S of natural numbers i such that theideal ai + md−1 corresponds to a point in Zd−1 . For i in S, the ideal ai + md corresponds
to a point in t−1
d (Zd−1 ), and so there is a closed subvariety Y of Hd (say, the closure of anirreducible component of this inverse image) which contains the point corresponding to
ai + md for infinitely many i in S. Let Zd be a subvariety of Y which is minimal amongclosed subvarieties that contain the point ai +md for infinitely many i in S; of course thisexists, since a descending chain of subvarieties of Hd has length at most the dimensionof Hd . In particular, these points ai + md are Zariski dense in Zd by this minimality.
By construction, td maps a dense open subset of Zd into Zd−1 , and this rational map isdominant by the minimality property of Zd−1 . Thus we have constructed Z0 , Z1 , Z2 , . . .
satisfying (1) and (2) by induction.
The dominant rational maps · · · 99K Z1 99K Z0 induce inclusions of function fields
k(Z0 ) ⊂ k(Z1 ) ⊂ · · · . Let K be the union of this sequence of fields. For each d ≥ 0,
the morphism Spec(K) → Zd ⊂ Hd corresponds to an ideal a0d in RK containing mdK .

1025–10

The compatibility between these morphisms implies that there is an ideal a, which is
unique, such that a0d = a + mdK for all d.
Definition 5.1. — An ideal a in K[[x1 , . . . , xn ]] as above is called a generic limit of
the sequence of ideals ai in k[[x1 , . . . , xn ]].
The following lemma shows how log canonical thresholds behave under the generic
limit construction. We refer to [DEM, Proposition 4.4] for the proof, which is elementary. The main point is that for a family of ideals parametrized for a variety Z, the log
canonical threshold is constant on some nonempty Zariski open subset of Z, and so ittakes that same value at the generic point of Z. This is immediate from the definitionof the log canonical threshold in terms of a log resolution.
Lemma 5.2. — Let a in RK = K[[x1 , . . . , xn ]] be a generic limit of the sequence ai of
ideals in k[[x1 , . . . , xn ]]. Assume that ai is contained in the maximal ideal mk for all i.
Let q be a rational number such that lct(a · mqK ) is computed by some divisor E whose
center in Spec RK is the origin. Let d0 be a natural number.
Then there is an integer d ≥ d0 and an infinite subset S of the natural numbers withthe following properties. First,
lct((a + mdK ) · mqK ) = lct((ai + mdk ) · mqk )for every i ∈ S. Furthermore, for every i ∈ S, lct((ai + mdk ) · mqk ) is computed by adivisor Ei whose center in Spec R is the origin, andordE (a + mdK ) = ordEi (ai + mdk ).
We deduce the following simple statement about the behavior of log canonicalthresholds under generic limits.
Corollary 5.3. — Let q be a nonnegative rational number. Then there is a strictlyincreasing sequence i0 , i1 , . . . of natural numbers such that lct(a · mqK ) = limj lct(aij · mqk ).
In particular, if the sequence lct(ai · mqk ) is convergent, then it converges to lct(a · mqK ).
Proof. We use a basic estimate for log canonical thresholds: for ideals c and c0 in
R = k[[x1 , . . . , xn ]] with c + mdk = c0 + mdk , we have|lct(c) − lct(c0 )| ≤nd[DM, Corollary 2.10]. Suppose inductively that we have chosen i0 < i1 < · · · < ij−1 .
By Lemma 5.2, there is an integer ij > ij−1 and an integer d ≥ j such that

1025–11

lct((a + mdK ) · mqK ) = lct((aij + mdk ) · mqk ). Therefore,|lct(a · mqK ) − lct(aid · mqk )| ≤ |lct(a · mqK ) − lct((a + mdK ) · mqK )|+ |lct((aid + mdk ) · mqk ) − lct(aid · mqk )|
2n
≤
d
2n
≤
.
j
The statement on limits follows immediately.

6. ACC FOR LOG CANONICAL THRESHOLDS ON SMOOTH
VARIETIESIn this section, we present the proof by de Fernex, Ein, and Mustaţă of the ACC
property for log canonical thresholds on smooth varieties [DEM]. We also prove thesmooth case of Kollár’s Accumulation Conjecture, proved by Kollár [KACC].
Theorem 6.1. — For each n ≥ 0, the set Tnsm ⊂ Q ∩ [0, n] satisfies the ascending
sm
chain condition, and its set of accumulation points is Tn−1.
A first easy step is to replace an ideal by another ideal with the same log canonicalthreshold, such that this log canonical threshold is computed by a divisor with a zerodimensional center, as follows (Lemma 5.2 in [DEM]).
Lemma 6.2. — Let a be an ideal contained in the maximal ideal mK of K[[x1 , . . . , xn ]].
Let q = max{t ≥ 0 : lct(a · mtK ) = lct(a)}.
(i) The nonnegative number q is rational.
(ii) We have lct(a · mqK ) = lct(a), and this log canonical threshold is computed by adivisor with center on X = Spec K[[x1 , . . . , xn ]] equal to the origin.
(iii) The number q is zero if and only if lct(a) is computed by a divisor with centeron X equal to the origin.
P
Proof. Let π : Y → X be a log resolution of a · mK , and write a · OY = OY (− ai Ei ),PPmK · OY = OY (− bi Ei ), and KY /X = KY ⊗ π ∗ (KX )∗ equal to OY ( i ei Ei ). Let I be
the set of those i such that Ei has center equal to the origin, that is, such that bi > 0.
Let c = lct(a). The log canonical threshold decreases (that is, the singularity gets
worse) as we increase a subscheme of X, and so lct(a·mtK ) ≤ c for all t ≥ 0. Furthermore,lct(a · mtK ) ≥ c if and only if
ei + 1 ≥ c(ai + tbi )for all i. If i is not in I, then bi = 0 and this inequality holds for all t. We concludethat
ei + 1 − cai:i∈I .
q = mincbi

1025–12

Thus q is rational. Moreover, there is an i ∈ I such that this minimum is achieved; thenEi computes lct(a · mqK ), and Ei has center on X equal to the origin. The statement in(iii) is clear.
Proof of Theorem 6.1. Let ci be a strictly increasing or strictly decreasing sequencein Tnsm . We will get a contradiction if the sequence is strictly increasing, while we willsm
show that the limit of the sequence is in Tn−1
if the sequence is strictly decreasing.
sm
Conversely, every number in Tn−1 is the limit of a strictly decreasing sequence in Tnsm .
Indeed, for an ideal a ⊂ k[[x1 , . . . , xn−1 ]], we have lct0 ((a, xdn )) = lct0 (a)+1/d by Lemma
3.2.
Let c = limi ci ; the limit is finite, since Tnsm is contained in [0, n]. For each i, there is
an ideal ai in R = k[[x1 , . . . , xn ]] with log canonical threshold at the origin equal to ci .
Let a be a generic limit of the sequence ai (Definition 5.1), with a ⊂ K[[x1 , . . . , xn ]]. By
construction of the generic limit, a is contained in the maximal ideal mK . By Corollary
5.3, the limiting ideal a has log canonical threshold equal to c. If c = 0, then thesequence ci must be strictly decreasing rather than strictly increasing. We must havesm
as we want. Thus we can assume that c > 0.
n > 0 (as T0sm = {0}), and so 0 is in Tn−1
In particular, the ideal a is not zero.
Let q be the rational number attached to a by Lemma 6.2. We havelct(a · mqK ) = lct(a).
We also have the trivial inequality
lct(ai · mqk ) ≤ lct(ai )for all i. In particular, when ci is a strictly increasing sequence, we have lct(ai · mqk ) <lct(a · mqK ) for all i.
By the choice of q, lct(a · mqK ) is computed by a divisor E which lies over the origin
in X = Spec K[[x1 , . . . , xn ]]. For any d > ordE (a · mqK ), the continuity lemma (Lemma
4.1) gives thatlct(a · mqK ) = lct((a + mdK ) · mq ),and E computes both log canonical thresholds.
By Lemma 5.2, there is an integer d > ordE (a · mqK ) and an infinite set S ⊂ N suchthat, for every i ∈ S, we have lct((a + mdK ) · mqK ) = lct((ai + mdk ) · mqk ). The lemma also
gives a divisor Ei over Spec k[[x1 , . . . , xn ]] which computes lct((ai + mdk ) · mqk ) and whichhas
ordEi ((ai + mdk ) · mqk ) = ordE ((a + mdK ) · mqK ) = ordE (a · mqK ).
By our inequality on d,
ordEi (mdk ) ≥ d > ordE (a · mqK ) = ordEi ((ai + mdk ) · mqk ).
By the continuity lemma (Lemma 4.1), for each i in the infinite set S, we havelct(ai · mqk ) = lct((ai + mdk ) · mqk ) = lct((a + mdK ) · mqK ) = lct(a · mqK ).

1025–13

If ci is a strictly increasing sequence, this equality contradicts the strict inequality
found earlier. Thus we have proved the ACC property for log canonical thresholds onsmooth varieties.
Finally, suppose that ci = lct(ai ) is a strictly decreasing sequence. We have shownthat the sequence lct(ai · mqk ) has infinitely many terms that are equal, and so q is not
zero. Equivalently, by Lemma 6.2, lct(a) is not computed by any divisor whose centerin X is the origin.Therefore, if F is a divisor over Spec K[[x1 , . . . , xn ]] that computes
the log canonical threshold of a, then the center of F in Spec K[[x1 , . . . , xn ]] has positive
dimension. After localizing at the generic point of this center, we deduce that lct(a) issm. (We use here that the completed local ring of a regular scheme containing a
in Tn−1
field at a codimension r integral subscheme W is isomorphic to the power series ringL[[x1 , . . . , xr ]], where L is the function field of W .) Since the numbers lct(ai ) convergesm.
to lct(a), we have shown that all accumulation points of Tnsm lie in Tn−1

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Burt TOTARODPMMSCambridge University
Wilberforce RoadCAMBRIDGE CB3 0WB
United Kingdom
E-mail : b.totaro@dpmms.cam.ac.uk